# NavList:

## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

**The Bygrave Slide Rule**

**From:**UNK

**Date:**2009 May 27, 14:57 -0700

I was asked recently to help a friend who wanted to understand the Bygrave Slide Rule (BSR). He sent me references to most of the internet postings, and Google helped me find more, including this forum. I have improvised such a device using scales printed on transparent sheets using computer programs I wrote for the purpose. A most interesting project. I am posting here now because I believe that some of the recent writings on the internet about the BSR may be misleading. I refer to the assertion by Gary LaPook (GLP) and Ron van Riet (RVR) that the inner scale of the BSR is actually a cotangent scale and not a tangent scale. The patent application, and the original manual of the BSR talk about tangents, so RVR and GLP seem to be suggesting that Bygrave may have made a mistake somewhere, even if only in the documentation. It seems quite reasonable to me that this scale can be described as a tangent scale. I think I can explain why this question arose in the first place. A conventional sliderule has a linear scale showing 1.0 at the left end and 10.0 at the right, the distance to a specific value from the left end being proportional to the logarithm of that value. The convention is clearly that the numbers (and their logarithms) increase from left to right along the scale. The inner scale of a BSR, if unwrapped horizontally, shows near-zero degrees at the left end and near-90 degrees at the right end, and the 45 degree mark is in the exact centre of the scale. We know that tangents of angles less than 45 degrees are less than one, and hence that logarithms of tangents of angles less than 45 degrees are negative. Similarly logs of tans of angles greater than 45 degrees are positive. The inner scale of the BSR, considered as a tangent scale, is oriented so that the logarithm of the tangent increases left-to-right in the conventional way. It IS reasonable to say this is a logarithmic tangent scale. So why are we discussing this at all? The point is that if BOTH scales were laid out in this way, the degree markings on the two scales would run in opposite directions. This arises since the tangent function INCREASES with increasing angle whereas the cosine function DECREASES. Several people have suggested that this would be prone to user error and Bygrave must have realised this. The BSR does have both degree scales running left-to-right. So how did Bygrave do it? Given that I have suggested that the inner scale is laid out in a reasonable way as a tangent scale, and that BOTH scales on the BSR have the degree markings left-to-right, I suggest that Bygrave achieved his objective by reversing the COSINE scale. The clever part is the way that Bygrave gets this reversed cosine scale to give the right answer. On a conventional slide rule, multiplication of X*Y is done by aligning the origen of the slider scale with X on the fixed scale, then reading off the product on the fixed scale opposite Y on the slider. The instructions with the BSR for the tan(X)/cos(Y) operation follow this precise procedure (think of inner=fixed and outer=sliding). Bygrave has cleverly transposed the multiply and divide instructions so that they give the right answer with the reversed log(cosine) scale. A reversed log(cosine) scale is the same as a log(1/cosine) scale so Bygrave is tricking us into multiplying by (1/cos(Y)) with the reversed cosine scale when he wants us to divide by cos(Y). A similar trick is used where Bygrave wants us to multiply by cosine, instructing us to perform the actions which we would normally use on a conventional slide rule to divide. The argument as to whether the inner scale is a tangent or a cotangent, (or whether the outer scale is a cosine or a secant) really comes down to whether you think of 'up' and 'left-to-right' as positive or negative. Bygrave used the terms 'tangent/cosine' in the theory, and 'inner/outer' in the instructions. There is no need for the end-user to know that one scale has been cleverly reversed so that he doesn't make mistakes reading off the degrees. regards Peter Martinez -------------------------------------------------------- [Sent from archive by: peter.martinez-AT-btinternet.com] --~--~---------~--~----~------------~-------~--~----~ Navigation List archive: www.fer3.com/arc To post, email NavList@fer3.com To , email NavList-@fer3.com -~----------~----~----~----~------~----~------~--~---